Dynamic Data Science Applications in Optimal Profit Algorithmic Trading

The file contains the source code of the data applications of the trading strategies presented in Dynamic Data Science Applications in Optimal Profit Algorithmic Trading.

Load required packages

pkg_list = c('quantmod', 'TTR', 'zoo', 'tseries', 'fGarch','PEIP','gdata',
             'gridExtra','tidyverse', 'aTSA', 'dygraphs', 'urca')
# Function to install required packages if needed
for (pkg in pkg_list)
{
  # Loading the library.
  if (!library(pkg, logical.return=TRUE, character.only=TRUE))
    {
         # If the library cannot be loaded, install first and then load.
        install.packages(pkg)
        library(pkg, character.only=TRUE)
  }
}

Load the required stocks

start.date = '2017-2-1' # starting date of stock
end.date = '2019-3-17' # ending date of stock
# Download the selected stocks from Yahoo finance
getSymbols(c('SPY','ADBE','EBAY','MSFT','IBM','GLD','GDX','EWA','EWC','IGE'), 
                      src = "yahoo", from = start.date, to = end.date)

##  [1] "SPY"  "ADBE" "EBAY" "MSFT" "IBM"  "GLD"  "GDX"  "EWA"  "EWC"  "IGE"

stocks <-merge(SPY = SPY[, "SPY.Adjusted"], ADBE = ADBE[, "ADBE.Adjusted"], 
               EBAY= EBAY[, "EBAY.Adjusted"], MSFT = MSFT[, "MSFT.Adjusted"], 
               IBM = IBM[, "IBM.Adjusted"], GLD = GLD[, "GLD.Adjusted"], 
               GDX = GDX[, "GDX.Adjusted"], EWA = EWA[, "EWA.Adjusted"], 
               EWC = EWC[, "EWC.Adjusted"], IGE = IGE[, "IGE.Adjusted"])
head(stocks)

##            SPY.Adjusted ADBE.Adjusted EBAY.Adjusted MSFT.Adjusted
## 2017-02-01     213.7879        113.36      31.55644      60.07822
## 2017-02-02     213.9288        113.16      31.35050      59.69079
## 2017-02-03     215.4034        115.17      31.44856      60.17270
## 2017-02-06     215.0183        114.46      31.40934      60.13491
## 2017-02-07     215.0277        114.96      31.80159      59.93647
## 2017-02-08     215.3095        116.13      32.60570      59.85144
##            IBM.Adjusted GLD.Adjusted GDX.Adjusted EWA.Adjusted
## 2017-02-01     149.5858       115.20     23.36890     18.50433
## 2017-02-02     149.8347       115.84     23.88821     18.69501
## 2017-02-03     150.8990       116.13     24.00579     18.71234
## 2017-02-06     150.9333       117.70     24.88764     18.53900
## 2017-02-07     153.1648       117.46     24.77986     18.49566
## 2017-02-08     152.3949       118.19     25.05421     18.59100
##            EWC.Adjusted IGE.Adjusted
## 2017-02-01     25.32722     31.71767
## 2017-02-02     25.39264     31.88734
## 2017-02-03     25.48610     32.13736
## 2017-02-06     25.30852     31.85162
## 2017-02-07     25.23376     31.38728
## 2017-02-08     25.32722     31.51230

Function to select cointegrated pairs

find_cointegrated_pairs <- function (data){
  n <- ncol (data)
  pvalue_matrix <- matrix(0, nrow=n, ncol=n)
  pairs <- list()
  m <- 1
  for (i in 1:n){
    for (j in 1:n){
      if(i>=j) { 
        next;
      } else{
      S1 <- data[, i]
      S2 <- data[, j]
      result <- coint.test (as.numeric(S1), as.numeric(S2), output = FALSE)
      pvalue_matrix [i, j] <- result[,3][[1]]
      if (result[,3][[1]] < 0.05){
        pairs [[m]] <- c (i, j)
        m <- m+1}
      }
    }
  }
  newlist <- list(pvalue_matrix, pairs)
  return (newlist)
}

Calculate p-value to identify cointegrated pairs

pvalue<-find_cointegrated_pairs(stocks)
#round(pvalue[[1]],3)

Plot the stock prices and test for multiple cointegration

assets <- c("EWC", "IGE") # selected two assets
pair.stock <- merge(stocks[, 9], stocks[, 10], join="inner")
colnames(pair.stock) <- assets
# Plot the assets 
plot(pair.stock, legend.loc=1)

# Test of multiple cointegration
jotest=ca.jo(pair.stock, type="trace", K=2, ecdet="none", spec="longrun")
summary(jotest)

## 
## ###################### 
## # Johansen-Procedure # 
## ###################### 
## 
## Test type: trace statistic , with linear trend 
## 
## Eigenvalues (lambda):
## [1] 0.013169621 0.009127074
## 
## Values of teststatistic and critical values of test:
## 
##           test 10pct  5pct  1pct
## r <= 1 |  4.87  6.50  8.18 11.65
## r = 0  | 11.91 15.66 17.95 23.52
## 
## Eigenvectors, normalised to first column:
## (These are the cointegration relations)
## 
##            EWC.l2     IGE.l2
## EWC.l2 1.00000000  1.0000000
## IGE.l2 0.01218259 -0.7776555
## 
## Weights W:
## (This is the loading matrix)
## 
##             EWC.l2     IGE.l2
## EWC.d -0.013956528 0.01218314
## IGE.d -0.005642251 0.02946432

Implementation of Non-Gaussian filter algorithm for two stocks

# select pairwise cointegrated stocks
x <- pair.stock[, 1]
y <- pair.stock[, 2]
x$intercept <- rep(1, nrow(x)) # create intercept
var_e <- 0.0001 # innovation covariance of observation
sigma_v <- var_e/(1-var_e)*diag(2) #covariance matrix of state
Ve <- 0.001
P_t <- 10^{-10}*diag(2)
P <- matrix(rep(0, 4), nrow=2)
I_t <- matrix(rep(0, 4), nrow=2) #information matrix
beta <- matrix(rep(0, nrow(y)*2), ncol=2)
y_fitted <- rep(0, nrow(y))
nu <- rep(0, nrow(y))
Q <- rep(0, nrow(y))
################################################################
# Function to implement the Non-Gaussian maximum filter operations
################################################################
kalman_iteration <- function(y, x) {
for(i in 1:nrow(y)) {
  if(i > 1) {
    beta[i, ] <- beta[i-1, ] # state transition
    P_t <- P + sigma_v # state covariance prediction
    }
  y_fitted[i] <- x[i, ] %*% beta[i, ] # observation prediction
  Q[i] <- x[i, ] %*% P_t %*% t(x[i, ]) + Ve # observation variance prediction
  nu[i] <- y[i] - y_fitted[i] # prediction error
  K_gain <- P_t %*% t(x[i, ]) / Q[i] # information gain
  # updating the state
  beta[i, ] <- beta[i, ] + K_gain * nu[i]
  I_t <- inv(P_t)+ t(x[i, ])%*%x[i, ] / Ve 
  P <- inv(I_t)
  
}
  return(list(beta, P, Q, nu))
}

res <- kalman_iteration(y,x) # Implementation of function
# Extract results
beta <- xts(res[[1]], order.by=index(pair.stock))
plot(beta[2:nrow(beta), 1],type='l',main ='Dynamic hedge ratio',col = "blue")

plot(beta[2:nrow(beta), 2],type='l',main ='Dynamic intercept',col = "blue")

# plot trade signals
nu <- xts(res[[4]], order.by=index(pair.stock))
sqrtQ <- xts(sqrt(res[[3]]), order.by=index(pair.stock))
signals <- merge(nu, sqrtQ, -sqrtQ)
colnames(signals) <- c("nu", "sqrtQ", "negsqrtQ")

Functions to calculate the optimal value of threshold \(p\), which maximizes the Sharpe ratio (SR) and cumulative profit

#############################################################
# Required functions to calculate p and profit
#############################################################
# Function to generate positions and calculate profit and loss for multiple stocks
PnL<-function(signals,nu,beta,x,y){
len <- length(index(signals)) 
vec.sig<-ifelse((signals[1:len]$nu > signals[1:len]$sqrtQ) & 
                  (lag.xts(signals$nu, 1) < lag.xts(signals$sqrtQ, 1)), -1, 
         ifelse((signals[1:len]$nu < signals[1:len]$negsqrtQ) & 
                 (lag.xts(signals$nu, 1) > lag.xts(signals$negsqrtQ, 1)), 1, 0))
colnames(vec.sig) <- "vectorise.signals"
# getting only the first signals
vec.sig[vec.sig == 0] <- NA # replace 0 by NA
vec.sig <- na.locf(vec.sig) # replace the missing values by last real observations
vec.sig <- diff(vec.sig)/2
# generate positions and calculate profit for two stocks
if(ncol(beta)==2){
sim <- merge(lag.xts(vec.sig,1), beta[, 1], x[, 1], y)
colnames(sim) <- c("sig", "hedge", assets[1], assets[2])
sim$posX <- sim$sig * -1000 * sim$hedge
sim$posY <- sim$sig * 1000   
sim$posX[sim$posX == 0] <- NA
sim$posX <- na.locf(sim$posX)
sim$posY[sim$posY == 0] <- NA
sim$posY <- na.locf(sim$posY)
PLX <- sim$posX * diff(sim[, assets[1]])
PLY <- sim$posY * diff(sim[, assets[2]])
profit_loss <- PLX + PLY
  }
# generate positions and calculate profit for three stocks
if(ncol(beta)==3){
sim <- merge(lag.xts(vec.sig,1), beta[, 1], beta[, 2], x[, 1], x[, 2], y)
colnames(sim) <- c("sig", "hedge1", "hedge2", assets[1], assets[2], assets[3])
sim$posX1 <- sim$sig * -1000 * sim$hedge1
sim$posX2 <- sim$sig * -1000 * sim$hedge2
sim$posY <- sim$sig * 1000   
sim$posX1[sim$posX1 == 0] <- NA
sim$posX1 <- na.locf(sim$posX1)
sim$posX2[sim$posX2 == 0] <- NA
sim$posX2 <- na.locf(sim$posX2)
sim$posY[sim$posY == 0] <- NA
sim$posY <- na.locf(sim$posY)
PLX <- sim$posX1 * diff(sim[, assets[1]]) + sim$posX2 * diff(sim[, assets[2]])
PLY <- sim$posY * diff(sim[, assets[3]])
profit_loss <- PLX + PLY
   }
# generate positions and calculate profit for four stocks
if(ncol(beta)==4){
sim <- merge(lag.xts(vec.sig,1), beta[,1], beta[,2], beta[,3], x[,1], x[,2], x[,3], y)
colnames(sim) <- c("sig", "hedge1", "hedge2", "hedge3",assets[1],assets[2], 
                                        assets[3], assets[4])
sim$posX1 <- sim$sig * -1000 * sim$hedge1
sim$posX2 <- sim$sig * -1000 * sim$hedge2
sim$posX3 <- sim$sig * -1000 * sim$hedge3
sim$posY <- sim$sig * 1000   
sim$posX1[sim$posX1 == 0] <- NA
sim$posX1 <- na.locf(sim$posX1)
sim$posX2[sim$posX2 == 0] <- NA
sim$posX2 <- na.locf(sim$posX2)
sim$posX3[sim$posX3 == 0] <- NA
sim$posX3 <- na.locf(sim$posX3)
sim$posY[sim$posY == 0] <- NA
sim$posY <- na.locf(sim$posY)
PLX <- sim$posX1 * diff(sim[, assets[1]]) + sim$posX2 * diff(sim[, assets[2]]) + 
                              sim$posX3 * diff(sim[, assets[3]])
PLY <- sim$posY * diff(sim[, assets[4]])
profit_loss <- PLX + PLY
    }
return(ProfitLoss=profit_loss)
}
# Functions to calculate the optimal value of threshold, $p$
SR.train<-function(nu, sqrtQ, p){
signals <- merge(nu, p*sqrtQ, -p*sqrtQ)
colnames(signals) <- c("nu", "sqrtQ", "negsqrtQ")
# Implementation of profit
profit.loss<- PnL(signals,nu,beta,x,y)
st_p <- sqrt(252)*mean(na.omit(profit.loss))/sd(na.omit(profit.loss))
return (st_p)
} 

# determining optimal p to maximize Sharpe ratio 
p <- seq(0.1, 2, 0.01)
SR<-0
for(j in 1:length(p)){
      SR[j] <- SR.train (nu, sqrtQ, p[j])
    }
max(na.omit(SR))

## [1] 1.370774

plot(p, SR, type = "l", col = "blue")

## Calcualte the optimal value of p by maximizing the SR

p.opt<-p[which.max(SR)]
p.opt

## [1] 1.34

# create optimal trading signals
p <- p.opt
signals <- merge(nu, p*sqrtQ, -p*sqrtQ)
colnames(signals) <- c("nu", "sqrtQ", "negsqrtQ")
plot(signals[3:length(index(signals))], ylab='nu', main = 'Trading signals', 
              col=c('blue', 'red', 'red'), lwd=c(1,2,2))

## Calculate the cumulative profit using optimal trading signals

# Implementation of profit and loss fucntion to calculate cumulative profit
profit.loss<- PnL(signals,nu,beta,x,y)
sum (na.omit(profit.loss))

## [1] 9340.675

plot(cumsum(na.omit(profit.loss)), main="Cumulative profit, $", col = "blue")

Implementation of Non-Gaussian filter for multiple trading (three stocks)

assets <- c("EWA", "EWC", "IGE")
three.stocks <- merge(stocks[, 8], stocks[, 9], stocks [, 10])
colnames(three.stocks) <- assets
plot(three.stocks, legend.loc=1)

jotest=ca.jo(three.stocks, type="trace", K=2, ecdet="none", spec="longrun")
summary(jotest)

## 
## ###################### 
## # Johansen-Procedure # 
## ###################### 
## 
## Test type: trace statistic , with linear trend 
## 
## Eigenvalues (lambda):
## [1] 0.038080520 0.011171405 0.009036416
## 
## Values of teststatistic and critical values of test:
## 
##           test 10pct  5pct  1pct
## r <= 2 |  4.82  6.50  8.18 11.65
## r <= 1 | 10.79 15.66 17.95 23.52
## r = 0  | 31.40 28.71 31.52 37.22
## 
## Eigenvectors, normalised to first column:
## (These are the cointegration relations)
## 
##             EWA.l2     EWC.l2    IGE.l2
## EWA.l2  1.00000000  1.0000000  1.000000
## EWC.l2 -0.47601178 -1.2951450 -3.893482
## IGE.l2 -0.01591207 -0.2048727  2.219207
## 
## Weights W:
## (This is the loading matrix)
## 
##            EWA.l2      EWC.l2       IGE.l2
## EWA.d -0.07058604 0.005843659 -0.003161974
## EWC.d -0.01404794 0.014292636 -0.003131786
## IGE.d  0.01849513 0.011053309 -0.009246670

Implementation of Non-Gaussian filter algorithm

x <- three.stocks[, 1:2]
y <- three.stocks[, 3]
x$intercept <- rep(1, nrow(x)) # create intercept
var_e <- 0.0001 # innovation covariance of observation
sigma_v <- var_e/(1-var_e)*diag(3) #covariance matrix of state
Ve <- 0.001
P_t <- 10^{-10}*diag(3)
P <- matrix(rep(0, 4), nrow=3)

## Warning in matrix(rep(0, 4), nrow = 3): data length [4] is not a sub-
## multiple or multiple of the number of rows [3]

I_t <- matrix(rep(0, 4), nrow=3) #information matrix

## Warning in matrix(rep(0, 4), nrow = 3): data length [4] is not a sub-
## multiple or multiple of the number of rows [3]

beta <- matrix(rep(0, nrow(y)*3), ncol=3)
y_fitted <- rep(0, nrow(y))
nu <- rep(0, nrow(y))
Q <- rep(0, nrow(y))
# Implementation of function of iterative Non-Gaussian filter operations
res <- kalman_iteration(y,x)
# Extract results
beta <- xts(res[[1]], order.by=index(three.stocks))
plot(beta[2:nrow(beta), 1:2], type='l', main = 'Dynamic hedge ratios', 
     col = c("blue", "green"))

plot(beta[2:nrow(beta), 3], type='l', main = 'Dynamic updated intercept', col = "blue")

# plot trade signals
nu <- xts(res[[4]], order.by=index(three.stocks))
sqrtQ <- xts(sqrt(res[[3]]), order.by=index(three.stocks))
signals <- merge(nu, sqrtQ, -sqrtQ)
colnames(signals) <- c("nu", "sqrtQ", "negsqrtQ")

# determining optimal p to maximize Sharpe ratio 
p <- seq(0.1, 2, 0.01)
SR<-0
for(j in 1:length(p)){
      SR[j] <- SR.train (nu, sqrtQ, p[j])
    }
max(na.omit(SR))

## [1] 1.581271

plot(p, SR, type = "l", col = "blue", ylab = "Annualized SR")

## Calcualte the optimal value of p by maximizing the SR

p.opt<-p[which.max(SR)]
p.opt

## [1] 1.02

# create optimal trading signals
p <- p.opt
signals <- merge(nu, p*sqrtQ, -p*sqrtQ)
colnames(signals) <- c("nu", "sqrtQ", "negsqrtQ")
plot(signals[3:length(index(signals))], ylab='nu', main = 'Trading signals', 
                col=c('blue', 'red', 'red'), lwd=c(1,2,2))

## Calculate the cumulative profit using optimal trading signals

# Implementation of profit and loss fucntion to calculate cumulative profit
profit.loss<- PnL(signals,nu,beta,x,y)
sum (na.omit(profit.loss))

## [1] 9713.534

plot(cumsum(na.omit(profit.loss)), main="Cumulative profit, $", col = "blue")

Implementation of Non-Gaussian filter for multiple trading (four stocks)

Plot and test for the presence of cointegration between stocks

assets <- c("GDX", "EWA", "EWC", "IGE")
four.stocks <- merge(stocks[, 7], stocks[, 8], stocks[, 9], stocks [, 10])
colnames(four.stocks) <- assets
plot(four.stocks, legend.loc=1)

jotest=ca.jo(four.stocks, type="trace", K=2, ecdet="none", spec="longrun")
summary(jotest)

## 
## ###################### 
## # Johansen-Procedure # 
## ###################### 
## 
## Test type: trace statistic , with linear trend 
## 
## Eigenvalues (lambda):
## [1] 0.045577162 0.021370707 0.011810312 0.009079026
## 
## Values of teststatistic and critical values of test:
## 
##           test 10pct  5pct  1pct
## r <= 3 |  4.84  6.50  8.18 11.65
## r <= 2 | 11.15 15.66 17.95 23.52
## r <= 1 | 22.62 28.71 31.52 37.22
## r = 0  | 47.39 45.23 48.28 55.43
## 
## Eigenvectors, normalised to first column:
## (These are the cointegration relations)
## 
##              GDX.l2     EWA.l2    EWC.l2     IGE.l2
## GDX.l2   1.00000000  1.0000000  1.000000   1.000000
## EWA.l2 -19.53896573 -0.4622338  3.276195   7.277618
## EWC.l2   9.13739689  0.5825642 -2.241764 -20.127084
## IGE.l2   0.02478951  0.2101317 -1.271165   8.400918
## 
## Weights W:
## (This is the loading matrix)
## 
##              GDX.l2        EWA.l2      EWC.l2        IGE.l2
## GDX.d  0.0026561203 -0.0252144712 0.001788170 -0.0005459360
## EWA.d  0.0040943496  0.0023533474 0.002981879 -0.0004305856
## EWC.d  0.0012926404 -0.0006573674 0.006025750 -0.0001939488
## IGE.d -0.0002119599  0.0011256165 0.007472117 -0.0016118821

Implementation of Non-Gaussian filter algorithm

x <- four.stocks[, 1:3]
y <- four.stocks[, 4]
x$intercept <- rep(1, nrow(x)) # create intercept
var_e <- 0.0001 # innovation covariance of observation
sigma_v <- var_e/(1-var_e)*diag(4) #covariance matrix of state
Ve <- 0.001
P_t <- 10^{-10}*diag(4)
P <- matrix(rep(0, 4), nrow=4)
I_t <- matrix(rep(0, 4), nrow=4) #information matrix
beta <- matrix(rep(0, nrow(y)*4), ncol=4)
y_fitted <- rep(0, nrow(y))
nu <- rep(0, nrow(y))
Q <- rep(0, nrow(y))
# Implementation of function of iterative Non-Gaussian filter operations
res <- kalman_iteration(y,x)
# Extract results
beta <- xts(res[[1]], order.by=index(four.stocks))
plot(beta[2:nrow(beta), 1:3], type='l', main = 'Dynamic updated hedge ratios', 
     col = c("blue", "green", "red"))

plot(beta[2:nrow(beta), 4], type='l', main = 'Dynamic updated intercept', col = "blue")

# plot trading signals
nu <- xts(res[[4]], order.by=index(four.stocks))
sqrtQ <- xts(sqrt(res[[3]]), order.by=index(four.stocks))
signals <- merge(nu, sqrtQ, -sqrtQ)
colnames(signals) <- c("nu", "sqrtQ", "negsqrtQ")

# determining optimal p to maximize Sharpe ratio 
p <- seq(0.1, 2, 0.01)
SR<-0
for(j in 1:length(p)){
      SR[j] <- SR.train (nu, sqrtQ, p[j])
    }
max(na.omit(SR))

## [1] 1.776355

plot(p, SR, type = "l", col = "blue", ylab = "Annualized SR")

## Calcualte the optimal value of p by maximizing the SR

p.opt<-p[which.max(SR)]
p.opt

## [1] 0.84

# create optimal trading signals
p <- p.opt
signals <- merge(nu, p*sqrtQ, -p*sqrtQ)
colnames(signals) <- c("nu", "sqrtQ", "negsqrtQ")
plot(signals[3:length(index(signals))], ylab='nu', main = 'Trading signals', 
                col=c('blue', 'red', 'red'), lwd=c(1,2,2))

## Calculate the cumulative profit using optimal trading signals

# Implementation of profit and loss fucntion to calculate cumulative profit
profit.loss<- PnL(signals,nu,beta,x,y)

plot(cumsum(na.omit(profit.loss)), main="Cumulative profit, $", col = "blue")

sum (na.omit(profit.loss))

## [1] 13334.45

Buy and hold strategy

maxx<- nrow (stocks)
GDX <- as.numeric(stocks [maxx, 7]) - as.numeric(stocks [1, 7])
EWA <- as.numeric(stocks [maxx, 8]) - as.numeric(stocks [1, 8])
EWC <- as.numeric(stocks [maxx, 9]) - as.numeric(stocks [1, 9])
IGE <- as.numeric(stocks [maxx, 10]) - as.numeric(stocks [1, 10])
1000*(EWC + IGE)

## [1] -778.936

1000*(EWA + EWC + IGE)

## [1] 1319.735

1000*(GDX + EWA + EWC + IGE)

## [1] 93.836